It is clearly explained in [1]} that in the minimal case (REF ) the on-shell description of the theory with two superfields with linearly realized supersymmetry coincides with [2]} and is equivalent to the VA non-linearly realized theory in eq. (REF ).
But the setup studied by Dall'Agata et al in [3]} is different from [1]} since they promote the Lagrange multiplier superfield \(T\)  to a propagating field, following [5]}. They add new terms in the Kähler potential
\(\Delta K = \beta \, T\bar{T} + g X\bar{X} T\bar{T} + {q\over 4} X\bar{X} X\bar{X} \ .\) 
